Estimation using LAFO Data

The estimated one-state VVM reproduces a qualitatively satisfactory output CZ(t) due to large amplitude change in angle-of-attack, but the accuracy is insufficient. The entire LAFO data available for CZ of GTA contains only a steady state oscillation cycle. Each CZ(t) is obtained by filtering the raw data over an appropriate bandwidth, and averaging

0.5 1 1.5 2 2.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 X: 1.785 Y: 0.2631 Frequency (Hz) PSD of C Z (t), dB/Hz

Fig. 4.9. Power Spectrum Density map ofCZ(t) of GTA in LAFO test with input α0 = 30o,∆α = 25o

andf = 1Hz.

the multiple oscillation cycles.

The estimation of model parameters from LAFO data was performed using a specialized software called ”Package for Interactive Identification” (PII), developed for this purpose by Goman and co-workers. The user manual of the software is in [91, 92], and case studies on identification of nonlinear unsteady aerodynamic model using this software are given in [93, 94]. A combination of the gradient and steepest-descent algorithms is used in the optimization process. The software provides a Graphical-User- Interface to check the quality of fit-obtained and alter the options of the optimization process.

Few samples of raw wind tunnel data from LAFO tests are available. As seen in PSD of CZ(t) presented in Fig.(4.9), there are significant peaks at first and second harmonics of input frequencies. This implies that the CZ variation for large amplitude inputs is nonlinear in nature. PSD of the steady state oscillation cycle for CZ shows a significant power only up to second harmonic frequency, for all the data. Hence, a two-state VVM is sufficient for modeling CZ.

Before estimation of the higher order kernel state equation parameters, the parameter functions (a(α), K1(α)) need to be defined. A simplistic approach is to define these parameter functions by using linear interpolation of the values of (a(α0), K1(α0)) for α _{∈ [35 : 5 : 55]. This definition is clearly insufficient for large amplitude simulations.} So, the model is updated with second kernel-state and used to estimate the parameters (K2(α), K3(α)) using LAFO data, to get an accurate model. However, it is observed that the contribution of x2(t) is small in comparison to x1(t), that is|x2(t)| < |x1(t)| . Hence,

20 25 30 35 40 45 50 −1.8 −1.6 −1.4 −1.2 −1 −0.8 Angle−of−attack α C Z (t) C mod C st C exp (a) α0= 35o, ∆α = 15o, f = 1Hz 10 20 30 40 50 −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 Angle−of−attack α C Z (t) C mod C st C exp (b) α0= 30o, ∆α = 20o, f = 0.5Hz

Fig. 4.10. Comparison of the output of VVM and wind tunnel test data, due to large amplitude oscillation input, forCZof GTA.

the role of second kernel state in this model is questionable.

It was observed that the mild nonlinearities in the variation of CZ can be captured by tuning of the nonlinear parameter functions (a(α), K1(α)) using LAFO data. In fact, the parameter functions are not defined completely as the values of (a(α0), K1(α0)) were estimated only for α _{∈ [35}o55o_{]. The modeling and measurement noises in SAFO data} are quantified by the error co-variance matrix. This co-variance matrix in turn gives the bounds on (a(α0), K1(α0). As shown in Fig.(4.7), the parameter bounds are a quiet significant percentage of the mean value. Hence, it is essential to tune (a(α), K1(α)) using the parameter bounds to account for the effects of noise as well as to estimate them over the extended range of α_{∈ [25}o60o].

So, the functions (a(α), K1(α)) are estimated considering these bounds as constraints and to produce a best fit with the LAFO data. The single-state VVM and Output-Error method are used in estimation of the parameter functions. An optimization is setup with root-mean-square-error between model output and experimental data as the cost function. Entire LAFO data is used simultaneously in the estimation process. It is found that the solution always converges to an approximately equivalent values of the parameter functions a(α) and K1(α). These are indicated by red solid lines, overlaid on the estimated parameter bounds from SAFO data, in Fig.(4.7). These estimated parameter functions are consistent with both SAFO and LAFO data.

The simulation results of the model thus obtained produce a good match with the LAFO data consistently. A sample of results of simulations is given in Fig.(4.10). Only the steady state oscillation cycle generated by the model is plotted in the figures. In these figures, Cexp indicates LAFO test data, Cst indicated the curve-fit of static test data, and Cmod indicates the model output. Same convention is followed throughout this chapter. The parameter function K1(α) is especially nonlinear near α = 40o. Also, the variations

5 10 15 20 25 30 35 40 45 50 55 Angle−of−attack α C Z (t) C Z, exp C Z, mod C Z, st C Z, raw

Fig. 4.11. Comparison of VVM output with raw wind tunnel test data forCZ of GTA, due to sinusoidal

input ofα0= 30o, ∆α = 25o, f = 1Hz.

at this angle-of-attack are very sensitive to external conditions. Hence, there is a minor inaccuracy in the model response at α = 40o. The break in oscillation cycle in Fig.(4.10,a) is seen when the angle-of-attack at last data point is not the same as first one. The rest of the results for LAFO data set are given in the APPENDIX C.

This model still produces minor inaccuracies at some angles-of-attack, with respect to the processed LAFO data. Consider Fig.(4.11), in which the model output is plotted with the raw wind tunnel data indicated by CZraw. It is evident that the single-state VVM

response is within the uncertainty bounds of the raw wind tunnel data, and hence the results are of acceptable accuracy. This also shows that, if the second kernel state is also included, it’s value is in the error tolerance limits, and hence it may not be realistic.

There are some higher frequency variations in CZraw(t) in Fig.(4.11) in the region

of sharp change in normal force coefficient. This can be observed in both pitch-up and pitch-down motion, but restricted to [40o₅₅o_{] range. The source of such higher frequency} local variations is unclear. Since the raw wind tunnel test data is severely limited, the consistency of this phenomenon cannot be confirmed either. It can be aerodynamic, or even due to external vibrations of the sting of test-rig. A better understanding of the test technique or flow dynamics is essential to resolve this matter. However, force and moment data are often filtered and processed using special techniques by the aerodynamicists. So,

these vibrations are not reflected in the processed LAFO data which are used in parameter estimation. Hence, the model output does not exhibit this phenomenon either.